Seminar on Differential Geometry. (AM-102), Volume 102 (Annals of Mathematics Studies, 102)

This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year 1979-1980; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program. Both survey articles and articles presenting new results are included. The articles on differential geometry and partial differential equations include a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincaré inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and metrics. The articles on differential geometry and mathematical physics cover such topics as renormalization, instantons, gauge fields and the Yang-Mills equation, nonlinear evolution equations, incompleteness of space-times, black holes, and quantum gravity. A feature of special interest is the inclusion of a list of more than one hundred unsolved research problems compiled by the editor with comments and bibliographical information. Cover......Page 1 Title Page......Page 4 Copyright Page......Page 5 TABLE OF CONTENTS......Page 6 INTRODUCTION......Page 10 SURVEY ON PARTIAL DIFFERENTIAL EQUATIONS IN DIFFERENTIAL GEOMETRY......Page 14 POINCARE INEQUALITIES ON RIEMANNIAN MANIFOLDS......Page 84 BONNESEN-TYPE INEQUALITIES IN ALGEBRAIC GEOMETRY, I. INTRODUCTION TO THE PROBLEM......Page 96 LIOUVILLE THEOREMS FOR HARMONIC MAPPINGS, AND AN APPROACH TO BERNSTEIN THEOREMS......Page 118 SUBHARMONIC FUNCTIONS, HARMONIC MAPPINGS AND ISOMETRIC IMMERSIONS......Page 144 AN ISOPERIMETRIC INEQUALITY AND WIEDERSEHEN MANIFOLDS......Page 154 ON THE BLASCHKE CONJECTURE......Page 170 BEST CONSTANTS IN THE SOBOLEV IMBEDDING THEOREM: THE YAMABE PROBLEM......Page 184 GAUSSIAN AND SCALAR CURVATURE, AN UPDATE......Page 196 CONFORMAL METRICS WITH ZERO SCALAR CURVATURE AND A SYMMETRIZATION PROCESS VIA MAXIMUM PRINCIPLE......Page 204 RIGIDITY OF POSITIVELY CURVED MANIFOLDS WITH LARGE DIAMETER......Page 214 COMPLETE THREE DIMENSIONAL MANIFOLDS WITH POSITIVE RICCI CURVATURE AND SCALAR CURVATURE......Page 220 ENTIRE SPACELIKE HYPERSURFACES OF CONSTANT MEAN CURVATURE IN MINKOWSKI SPACE......Page 240 APPLICATIONS OF THE MONGE-AMPERE OPERATORS TO THE DIRICHLET PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS......Page 250 EXTREMAL KAHLER METRICS......Page 270 L^2 HARMONIC FORMS ON COMPLETE MANIFOLDS......Page 302 L^2-COHOMOLOGY AND INTERSECTION HOMOLOGY OF SINGULAR ALGEBRAIC VARIETIES......Page 314 FUNCTION THEORY OF NONCOMPACT KAHLER MANIFOLDS OF NONPOSITIVE CURVATURE......Page 352 KAHLER MANIFOLDS WITH VANISHING FIRST CHERN CLASS......Page 370 COMPACTIFICATION OF NEGATIVELY CURVED COMPLETE KAHLER MANIFOLDS OF FINITE VOLUME......Page 374 LOCAL ISOMETRIC EMBEDDINGS......Page 392 YANG-MILLS THEORY: ITS PHYSICAL ORIGINS AND DIFFERENTIAL GEOMETRIC ASPECTS......Page 406 SYMMETRY AND ISOLATED SINGULARITIES OF CONFORMALLY FLAT METRICS AND OF SOLUTIONS OF THE YANG-MILLS EQUATIONS......Page 434 ON PARALLEL YANG-MILLS FIELDS......Page 454 VARIATIONAL PROBLEMS FOR GAUGE FIELDS......Page 466 SOME GEOMETRICAL ASPECTS OF INTEGRABLE NONLINEAR EVOLUTION EQUATIONS......Page 476 THE CONFORMALLY INVARIANT LAPLACIAN AND THE INSTANTON VANISHING THEOREM......Page 494 CAUSALLY DISCONNECTING SETS, MAXIMAL GEODESICS AND GEODESIC INCOMPLETENESS FOR STRONGLY CAUSAL SPACE-TIMES......Page 510 RENORMALIZATION......Page 518 METRICS WITH PRESCRIBED RICCI CURVATURE......Page 536 BLACK HOLE UNIQUENESS THEOREMS IN CLASSICAL AND QUANTUM GRAVITY......Page 550 GRAVITATIONAL INSTANTONS......Page 614 SOME UNSOLVED PROBLEMS IN CLASSICAL GENERAL RELATIVITY......Page 642 PROBLEM SECTION......Page 680 Edited By Shing-tung Yau. Collection Of Papers Presented At Seminars In The Academic Year 1979-80 Sponsored By The Institute For Advanced Study And The National Science Foundation. Includes Bibliographies.